On properties of adjoint polynomials of graphs and their applications

For a graph G, we denote by P(G, λ) the chromatic polynomial of G and by h(G, x) the adjoint polynomial of G. A graph G is said to be chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H ≅ G. In this paper, we investigate some algebraic properties of the adjoint polynomials of some graphs. Using these properties, we obtain necessary and sufficient conditions for K_n - E(∪_a,bT_1,a,b) and (∪_iC_ni) ∪ (∪_iD_mj) ∪ (∪_a,bT_1,a,b) to be chromatically unique if G_i ∈ {C_n,D_n, T_1,a,b|n ≥ 5, 3 ≤ a ≤ 10, a ≤ b} and h(P_m) h(G_i) for all m ≥ 2. Moreover, many new chromatically unique graphs are given.